The Boundary of a Square Tiling of a Graph Coincides with the Poisson Boundary
The Boundary of a Square Tiling of a Graph coincides with the Poisson Boundary Agelos Georgakopoulos∗ Mathematics Institute University of Warwick CV4 7AL, UK November 21, 2012 Abstract Answering a question of Benjamini & Schramm [4], we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, namely as the boundary of a square tiling of a cylinder. For this, we introduce a general criterion for identifying the Poisson boundary of a Markov chain that might have further applications. 1 Introduction 1.1 Overview In this paper we prove the following fact, conjectured by Benjamini & Schramm [4, Question 7.4.] Theorem 1.1. If G is a plane, uniquely absorbing graph with bounded degrees, then the boundary of its square tiling is a realisation of its Poisson boundary. The main implication of this is that every such graph G can be embedded inside the unit disc D of the real plane in such a way that random walk on G converges to ∂ D almost surely, its exit distribution coincides with Lebesgue measure on ∂ D, and there is a one-to-one correspondence between the bounded ∞ harmonic functions on G and L (∂ D) (the innovation of Theorem 1.1 is the last sentence). 2 A plane graph G ⊂ Ê is uniquely absorbing, if for every finite subgraph G0 2 there is exactly one connected component D of Ê \G0 that is absorbing, that is, random walk on G visits G\D only finitely many times with positive probability (in particular, G is transient).
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